The controversial summit included a closed-door, one-on-one meeting between the two leaders that left some high-level Trump administration intelligence officials in the dark.
At the annual Aspen Security Forum in Colorado days after the summit, U.S. intelligence chief Dan Coats said he had not been informed about the lengthy behind-the-scenes meeting.
"If he had asked me how that ought to be conducted, I would have suggested a different way, but that's not my role," Coats said. "It is what it is."
Mattis' remarks to Pentagon reporters on Friday followed a series of statements from the White House and the Kremlin that made separate claims about the agreements hashed out during that discussion.
Trump said he and Putin had discussed ways to work together to curb nuclear weapons in North Korea. "Russia has assured us of its support," the president said, adding that "discussions are ongoing and they're going very, very well."
But one day after the summit, the Russian Defense Ministry announced it was "ready" to put in place the "international security" measures agreed upon during the closed-door meeting.
At a news conference following the one-on-one, Trump said Putin had made an "incredible offer" for members of special counsel Robert Mueller's probe of Russian election meddling to "come and work" with Kremlin investigators. The White House later rejected Putin's offer.
On Friday, Mattis confirmed earlier reports that he was considering the possibility of talks with his Russian counterpart, Defense Minister Sergey Shoygu, signaling further willingness to establish regular relations between the two adversarial countries.
He also told reporters to "rest assured there are actions underway" by the U.S. "to protect our elections or to expose any external any, by anybody — external efforts to influence the American public."
Mattis did not provide any details to reporters about those efforts.
Minutes after Mattis' remarks, however, the White House said, Trump "looks forward to having President Putin to Washington after the first of the year, and he is open to visiting Moscow upon receiving a formal invitation."
— CNBC's Dan Mangan contributed to this report.